Incidences and the spectra of graphs

Abstract

In this paper we give incidence bounds for arrangements of curves in F 2 q. As an application, we prove a new result that if (x, f (x)) is a Sidon set then either A + A or f (A) + f (A) should be large. The main goal of the paper is to illustrate the use of graph spectral techniques in additive combinatorics. This is an extended version of the talks I gave in the Additive Combinatorics DocCourse held at the CRM in Barcelona and at the conference "Fete of Combinatorics" held in Keszthely.